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Theorem unineq 3505
 Description: Infer equality from equalities of union and intersection. Exercise 20 of [Enderton] p. 32 and its converse. (Contributed by NM, 10-Aug-2004.)
Assertion
Ref Expression
unineq (((AC) = (BC) (AC) = (BC)) ↔ A = B)

Proof of Theorem unineq
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 eleq2 2414 . . . . . . 7 ((AC) = (BC) → (x (AC) ↔ x (BC)))
2 elin 3219 . . . . . . 7 (x (AC) ↔ (x A x C))
3 elin 3219 . . . . . . 7 (x (BC) ↔ (x B x C))
41, 2, 33bitr3g 278 . . . . . 6 ((AC) = (BC) → ((x A x C) ↔ (x B x C)))
5 iba 489 . . . . . . 7 (x C → (x A ↔ (x A x C)))
6 iba 489 . . . . . . 7 (x C → (x B ↔ (x B x C)))
75, 6bibi12d 312 . . . . . 6 (x C → ((x Ax B) ↔ ((x A x C) ↔ (x B x C))))
84, 7syl5ibr 212 . . . . 5 (x C → ((AC) = (BC) → (x Ax B)))
98adantld 453 . . . 4 (x C → (((AC) = (BC) (AC) = (BC)) → (x Ax B)))
10 uncom 3408 . . . . . . . . 9 (AC) = (CA)
11 uncom 3408 . . . . . . . . 9 (BC) = (CB)
1210, 11eqeq12i 2366 . . . . . . . 8 ((AC) = (BC) ↔ (CA) = (CB))
13 eleq2 2414 . . . . . . . 8 ((CA) = (CB) → (x (CA) ↔ x (CB)))
1412, 13sylbi 187 . . . . . . 7 ((AC) = (BC) → (x (CA) ↔ x (CB)))
15 elun 3220 . . . . . . 7 (x (CA) ↔ (x C x A))
16 elun 3220 . . . . . . 7 (x (CB) ↔ (x C x B))
1714, 15, 163bitr3g 278 . . . . . 6 ((AC) = (BC) → ((x C x A) ↔ (x C x B)))
18 biorf 394 . . . . . . 7 x C → (x A ↔ (x C x A)))
19 biorf 394 . . . . . . 7 x C → (x B ↔ (x C x B)))
2018, 19bibi12d 312 . . . . . 6 x C → ((x Ax B) ↔ ((x C x A) ↔ (x C x B))))
2117, 20syl5ibr 212 . . . . 5 x C → ((AC) = (BC) → (x Ax B)))
2221adantrd 454 . . . 4 x C → (((AC) = (BC) (AC) = (BC)) → (x Ax B)))
239, 22pm2.61i 156 . . 3 (((AC) = (BC) (AC) = (BC)) → (x Ax B))
2423eqrdv 2351 . 2 (((AC) = (BC) (AC) = (BC)) → A = B)
25 uneq1 3411 . . 3 (A = B → (AC) = (BC))
26 ineq1 3450 . . 3 (A = B → (AC) = (BC))
2725, 26jca 518 . 2 (A = B → ((AC) = (BC) (AC) = (BC)))
2824, 27impbii 180 1 (((AC) = (BC) (AC) = (BC)) ↔ A = B)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ∪ cun 3207   ∩ cin 3208 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214 This theorem is referenced by:  phiall  4618
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